MATH.20:4.6 - Use the bounded result and stop at the needed strength

Return the comparator, its relation to the requested result and the premises needed by the next use. A numerical interval, a set inclusion or a short inequality can carry this result.

For a minimization with feasible value U and lower bound L, the candidate is within U-L of the optimal value when these quantities are finite. The receiver can use that gap to decide whether further optimization matters. If only exclusion, existence or a threshold result was needed, stop once that result follows.

General choices about cost, value and further work use FPF’s existing characterization, Pareto and improvement methods. Componentwise mathematical bounds can inform them, but a candidate attaining one coordinate need not attain another. Preserve which constructions actually realize the compared results.

Reopen the affected bound when its domain, objective, available information or following operation changes. Preserve bounds whose proofs still apply.